OCR C3 (Core Mathematics 3) 2010 June

1 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in each of the following cases:

1. \(y = x ^ { 3 } \mathrm { e } ^ { 2 x }\),
2. \(y = \ln \left( 3 + 2 x ^ { 2 } \right)\),
3. \(y = \frac { x } { 2 x + 1 }\).

2 The transformations R, S and T are defined as follows.
R : reflection in the \(x\)-axis
S : stretch in the \(x\)-direction with scale factor 3
\(\mathrm { T } : \quad\) translation in the positive \(x\)-direction by 4 units

1. The curve \(y = \ln x\) is transformed by R followed by T . Find the equation of the resulting curve.
2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }\). You should make clear the order of the transformations.

3

1. Express the equation \(\operatorname { cosec } \theta ( 3 \cos 2 \theta + 7 ) + 11 = 0\) in the form \(a \sin ^ { 2 } \theta + b \sin \theta + c = 0\), where \(a , b\) and \(c\) are constants.
2. Hence solve, for \(- 180 ^ { \circ } < \theta < 180 ^ { \circ }\), the equation \(\operatorname { cosec } \theta ( 3 \cos 2 \theta + 7 ) + 11 = 0\).
   \includegraphics[max width=\textwidth, alt={}, center]{cd1bde44-ab7e-45e6-ac22-346145eba3a0-2_648_951_1530_598} The diagram shows part of the curve \(y = \frac { k } { x }\), where \(k\) is a positive constant. The points \(A\) and \(B\) on the curve have \(x\)-coordinates 2 and 6 respectively. Lines through \(A\) and \(B\) parallel to the axes as shown meet at the point \(C\). The region \(R\) is bounded by the curve and the lines \(x = 2 , x = 6\) and \(y = 0\). The region \(S\) is bounded by the curve and the lines \(A C\) and \(B C\). It is given that the area of the region \(R\) is \(\ln 81\).
3. Show that \(k = 4\).
4. Find the exact volume of the solid produced when the region \(S\) is rotated completely about the \(x\)-axis.
5. Solve the inequality \(| 2 x + 1 | \leqslant | x - 3 |\).
6. Given that \(x\) satisfies the inequality \(| 2 x + 1 | \leqslant | x - 3 |\), find the greatest possible value of \(| x + 2 |\).
7. Show by calculation that the equation $$\tan ^ { 2 } x - x - 2 = 0$$ where \(x\) is measured in radians, has a root between 1.0 and 1.1.
8. Use the iteration formula \(x _ { n + 1 } = \tan ^ { - 1 } \sqrt { 2 + x _ { n } }\) with a suitable starting value to find this root correct to 5 decimal places. You should show the outcome of each step of the process.
9. Deduce a root of the equation $$\sec ^ { 2 } 2 x - 2 x - 3 = 0$$
   \includegraphics[max width=\textwidth, alt={}]{cd1bde44-ab7e-45e6-ac22-346145eba3a0-3_771_1087_1128_529}
   The diagram shows the curve with equation \(y = ( 3 x - 1 ) ^ { 4 }\). The point \(P\) on the curve has coordinates \(( 1,16 )\) and the tangent to the curve at \(P\) meets the \(x\)-axis at the point \(Q\). The shaded region is bounded by \(P Q\), the \(x\)-axis and that part of the curve for which \(\frac { 1 } { 3 } \leqslant x \leqslant 1\). Find the exact area of this shaded region.
10. Express \(3 \cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
11. The expression \(\mathrm { T } ( x )\) is defined by \(\mathrm { T } ( x ) = \frac { 8 } { 3 \cos x + 3 \sin x }\).
    (a) Determine a value of \(x\) for which \(\mathrm { T } ( x )\) is not defined.
    (b) Find the smallest positive value of \(x\) satisfying \(\mathrm { T } ( 3 x ) = \frac { 8 } { 9 } \sqrt { 6 }\), giving your answer in an exact form. \section*{[Question 9 is printed overleaf.]}

9 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = 4 x ^ { 2 } - 12 x \quad \text { and } \quad \mathrm { g } ( x ) = a x + b$$ where \(a\) and \(b\) are non-zero constants.

1. Find the range of f .
2. Explain why the function \(f\) has no inverse.
3. Given that \(\mathrm { g } ^ { - 1 } ( x ) = \mathrm { g } ( x )\) for all values of \(x\), show that \(a = - 1\).
4. Given further that \(\operatorname { gf } ( x ) < 5\) for all values of \(x\), find the set of possible values of \(b\).